1. Field of the Invention
This invention relates to optical signal processing, and more particularly to the use of fractional Fourier transform properties of lenses to correct the effects of lens misfocus in photographs, video, and other types of captured images.
2. Discussion of the Related Art
A number of references are cited herein; these are provided in a numbered list at the end of the detailed description of the preferred embodiments. These references are cited at various locations throughout the specification using a reference number enclosed in square brackets.
The Fourier transforming properties of simple lenses and related optical elements is well known and heavily used in a branch of engineering known as Fourier optics [1, 2]. Classical Fourier optics [1, 2, 3, 4] utilize lenses or other means to obtain a two-dimensional Fourier transform of an optical wavefront, thus creating a Fourier plane at a particular spatial location relative to an associated lens. This Fourier plane includes an amplitude distribution of an original two-dimensional optical image, which becomes the two-dimensional Fourier transform of itself. In the far simpler area of classical geometric optics [1, 3], lenses and related objects are used to change the magnification of a two-dimensional image according to the geometric relationship of the classical lens-law. It has been shown that between the geometries required for classical Fourier optics and classical geometric optics, the action of a lens or related object acts on the amplitude distribution of images as the fractional power of the two-dimensional Fourier transform. The fractional power of the fractional Fourier transform is determined by the focal length characteristics of the lens, and the relative spatial separation between a lens, source image, and an observed image.
The fractional Fourier transform has been independently discovered on various occasions over the years [5, 7, 8, 9, 10], and is related to several types of mathematical objects such as the Bargmann transform [8] and the Hermite semigroup [13]. As shown in [5], the most general form of optical properties of lenses and other related elements [1, 2, 3] can be transformed into a fractional Fourier transform representation. This property has apparently been rediscovered some years later and worked on steadily ever since (see for example [6]), expanding the number of optical elements and situations covered. It is important to remark, however, that the lens modeling approach in the latter ongoing series of papers view the multiplicative phase term in the true form of the fractional Fourier transform as a problem or annoyance and usually omit it from consideration.